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Properties

Label 1-1287-1287.761-r1-0-0
Degree $1$
Conductor $1287$
Sign $0.338 + 0.940i$
Analytic cond. $138.307$
Root an. cond. $138.307$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.587 − 0.809i)2-s + (−0.309 − 0.951i)4-s + (0.406 − 0.913i)5-s + (0.743 + 0.669i)7-s + (−0.951 − 0.309i)8-s + (−0.5 − 0.866i)10-s + (0.978 − 0.207i)14-s + (−0.809 + 0.587i)16-s + (0.104 + 0.994i)17-s + (0.743 − 0.669i)19-s + (−0.994 − 0.104i)20-s + (−0.5 + 0.866i)23-s + (−0.669 − 0.743i)25-s + (0.406 − 0.913i)28-s + (0.309 + 0.951i)29-s + ⋯
L(s)  = 1  + (0.587 − 0.809i)2-s + (−0.309 − 0.951i)4-s + (0.406 − 0.913i)5-s + (0.743 + 0.669i)7-s + (−0.951 − 0.309i)8-s + (−0.5 − 0.866i)10-s + (0.978 − 0.207i)14-s + (−0.809 + 0.587i)16-s + (0.104 + 0.994i)17-s + (0.743 − 0.669i)19-s + (−0.994 − 0.104i)20-s + (−0.5 + 0.866i)23-s + (−0.669 − 0.743i)25-s + (0.406 − 0.913i)28-s + (0.309 + 0.951i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.338 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.338 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $0.338 + 0.940i$
Analytic conductor: \(138.307\)
Root analytic conductor: \(138.307\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (761, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (1:\ ),\ 0.338 + 0.940i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4844729699 + 0.3406068492i\)
\(L(\frac12)\) \(\approx\) \(0.4844729699 + 0.3406068492i\)
\(L(1)\) \(\approx\) \(1.115368296 - 0.6279699576i\)
\(L(1)\) \(\approx\) \(1.115368296 - 0.6279699576i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.587 - 0.809i)T \)
5 \( 1 + (0.406 - 0.913i)T \)
7 \( 1 + (0.743 + 0.669i)T \)
17 \( 1 + (0.104 + 0.994i)T \)
19 \( 1 + (0.743 - 0.669i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.309 + 0.951i)T \)
31 \( 1 + (-0.994 - 0.104i)T \)
37 \( 1 + (-0.743 - 0.669i)T \)
41 \( 1 + (-0.743 + 0.669i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.743 + 0.669i)T \)
53 \( 1 + (0.809 + 0.587i)T \)
59 \( 1 + (-0.951 + 0.309i)T \)
61 \( 1 + (-0.913 - 0.406i)T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 + (-0.994 + 0.104i)T \)
73 \( 1 + (-0.951 + 0.309i)T \)
79 \( 1 + (-0.913 + 0.406i)T \)
83 \( 1 + (0.994 - 0.104i)T \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 + (-0.994 - 0.104i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.79818379356340223560903910392, −20.225967773747763833633018004447, −18.84444856572381067535100675844, −18.11882466683862577093880770966, −17.674790418127700615304485964640, −16.714042710708997549777591364645, −16.13125526395290700342862765132, −15.02878193492953494972898290167, −14.55644230056158631141194486390, −13.77401331666537851725435908622, −13.44589846535764566543785666635, −12.07251794168809303142827484757, −11.54315166630250333168594135329, −10.48705664361144026037415327051, −9.7436115537826861572799642094, −8.60126116454720270163164447005, −7.69743325025632986409621261160, −7.15167771355425346466086845060, −6.33750404717930813052148475861, −5.44507523985822510653026428387, −4.63785081230755075951751536805, −3.6614082315749481799968736224, −2.855164760044122626385846769267, −1.70371772542305528331756617587, −0.08286125905475079957726622016, 1.39940159778730074386593931454, 1.745795429761824340197041688236, 2.94537023014858127932805843343, 3.97460627884174671719342203475, 4.94537077535763583143842728388, 5.431410769367872671205707957750, 6.208004187220246068326724360635, 7.6015538245456422281718813497, 8.73687334535535047881437715628, 9.13882701147931380400001341713, 10.128940166531873740683950122187, 10.96386469892277600959683579204, 11.861946471308472637638002970, 12.345331503095402425885428454467, 13.19434309876482570001777430557, 13.85361476716477935132205849829, 14.6949852033080032515561410581, 15.4307127559734647947361549439, 16.2680431334202807734940957439, 17.416332066684060329606111213174, 17.96058019158898466295441778898, 18.749410572737028362001219995642, 19.82619995504580950787922882754, 20.16377850818675926444870469622, 21.11203262909420246824777058528

Graph of the $Z$-function along the critical line